Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A

Percentage Accurate: 99.9% → 99.9%

Time: 13.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))

C

double code(double x, double y, double z, double t) {
	return (((x * log(y)) - y) - z) + log(t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * log(y)) - y) - z) + log(t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * Math.log(y)) - y) - z) + Math.log(t);
}

Python

def code(x, y, z, t):
	return (((x * math.log(y)) - y) - z) + math.log(t)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * log(y)) - y) - z) + log(t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot \log y - y\right) - z\right) + \log t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (- (- (* x (log y)) y) z) (log t)))

C

double code(double x, double y, double z, double t) {
	return (((x * log(y)) - y) - z) + log(t);
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * log(y)) - y) - z) + log(t)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * Math.log(y)) - y) - z) + Math.log(t);
}

Python

def code(x, y, z, t):
	return (((x * math.log(y)) - y) - z) + math.log(t)

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * log(y)) - y) - z) + log(t))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * log(y)) - y) - z) + log(t);
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] - z), $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (fma x (log y) (log t)) (+ y z)))

C

double code(double x, double y, double z, double t) {
	return fma(x, log(y), log(t)) - (y + z);
}

Julia

function code(x, y, z, t)
	return Float64(fma(x, log(y), log(t)) - Float64(y + z))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x * N[Log[y], $MachinePrecision] + N[Log[t], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\log t + \left(\left(x \cdot \log y - y\right) - z\right)} \]

   3. lift--.f64 N/A

      \[\leadsto \log t + \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} \]

   4. lift--.f64 N/A

      \[\leadsto \log t + \left(\color{blue}{\left(x \cdot \log y - y\right)} - z\right) \]

   5. associate--l- N/A

      \[\leadsto \log t + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} \]

   6. associate-+r- N/A

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

   8. +-commutative N/A

      \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} - \left(y + z\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - \left(y + z\right) \]

   11. lower-+.f64 99.8

       \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \color{blue}{\left(y + z\right)} \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)} \]
5. Add Preprocessing

Alternative 2: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ t_2 := t\_1 - y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+223}:\\ \;\;\;\;-y\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-y}{z}, -z\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\log t - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))) (t_2 (- t_1 y)))
   (if (<= t_2 -5e+223)
     (- y)
     (if (<= t_2 -2e+14)
       (fma z (/ (- y) z) (- z))
       (if (<= t_2 2e-9) (- (log t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double t_2 = t_1 - y;
	double tmp;
	if (t_2 <= -5e+223) {
		tmp = -y;
	} else if (t_2 <= -2e+14) {
		tmp = fma(z, (-y / z), -z);
	} else if (t_2 <= 2e-9) {
		tmp = log(t) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	t_2 = Float64(t_1 - y)
	tmp = 0.0
	if (t_2 <= -5e+223)
		tmp = Float64(-y);
	elseif (t_2 <= -2e+14)
		tmp = fma(z, Float64(Float64(-y) / z), Float64(-z));
	elseif (t_2 <= 2e-9)
		tmp = Float64(log(t) - z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+223], (-y), If[LessEqual[t$95$2, -2e+14], N[(z * N[((-y) / z), $MachinePrecision] + (-z)), $MachinePrecision], If[LessEqual[t$95$2, 2e-9], N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 x (log.f64 y)) y) < -4.99999999999999985e223

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 60.4

         \[\leadsto \color{blue}{-y} \]

   5. Applied rewrites 60.4%

      \[\leadsto \color{blue}{-y} \]

   if -4.99999999999999985e223 < (-.f64 (*.f64 x (log.f64 y)) y) < -2e14

   1. Initial program 100.0%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\left(x \cdot \log y - y\right) - z\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \log t + \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \log t + \left(\color{blue}{\left(x \cdot \log y - y\right)} - z\right) \]

      5. associate--l- N/A

         \[\leadsto \log t + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} \]

      6. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} - \left(y + z\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - \left(y + z\right) \]

      11. lower-+.f64 100.0

          \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \color{blue}{\left(y + z\right)} \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z} + 1\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)\right)} + \left(\mathsf{neg}\left(z\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \color{blue}{z \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}} + \left(\mathsf{neg}\left(z\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\left(\log t + x \cdot \log y\right) - y}{z}, \mathsf{neg}\left(z\right)\right)} \]

   7. Applied rewrites 86.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(x, \log y, \log t\right) - y}{z}, -z\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{-1 \cdot y}{z}, \mathsf{neg}\left(z\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.1%

       \[\leadsto \mathsf{fma}\left(z, \frac{-y}{z}, -z\right) \]

   if -2e14 < (-.f64 (*.f64 x (log.f64 y)) y) < 2.00000000000000012e-9

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log t} - \left(y + z\right) \]

      3. lower-+.f64 98.4

         \[\leadsto \log t - \color{blue}{\left(y + z\right)} \]

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

   6. Taylor expanded in y around 0

      \[\leadsto \log t - \color{blue}{z} \]
   7. Step-by-step derivation

   8. Applied rewrites 96.6%

      \[\leadsto \log t - \color{blue}{z} \]

   if 2.00000000000000012e-9 < (-.f64 (*.f64 x (log.f64 y)) y)

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} \]

      2. lower-log.f64 84.6

         \[\leadsto x \cdot \color{blue}{\log y} \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{x \cdot \log y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 150:\\ \;\;\;\;\mathsf{fma}\left(x, \log y, \log t - z\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \log y - \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 150.0) (fma x (log y) (- (log t) z)) (- (* x (log y)) (+ y z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 150.0) {
		tmp = fma(x, log(y), (log(t) - z));
	} else {
		tmp = (x * log(y)) - (y + z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 150.0)
		tmp = fma(x, log(y), Float64(log(t) - z));
	else
		tmp = Float64(Float64(x * log(y)) - Float64(y + z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 150.0], N[(x * N[Log[y], $MachinePrecision] + N[(N[Log[t], $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 150

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - z} \]
   4. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) + \left(\mathsf{neg}\left(z\right)\right)} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} + \left(\mathsf{neg}\left(z\right)\right) \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{x \cdot \log y + \left(\log t + \left(\mathsf{neg}\left(z\right)\right)\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t + \left(\mathsf{neg}\left(z\right)\right)\right)} \]

      5. lower-log.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{\log y}, \log t + \left(\mathsf{neg}\left(z\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\log t - z}\right) \]

      7. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\log t - z}\right) \]

      8. lower-log.f64 98.7

         \[\leadsto \mathsf{fma}\left(x, \log y, \color{blue}{\log t} - z\right) \]

   5. Applied rewrites 98.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t - z\right)} \]

   if 150 < y

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\left(x \cdot \log y - y\right) - z\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \log t + \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \log t + \left(\color{blue}{\left(x \cdot \log y - y\right)} - z\right) \]

      5. associate--l- N/A

         \[\leadsto \log t + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} \]

      6. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} - \left(y + z\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - \left(y + z\right) \]

      11. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \color{blue}{\left(y + z\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]

      2. lower-log.f64 99.7

         \[\leadsto x \cdot \color{blue}{\log y} - \left(y + z\right) \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.6 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \log y - \left(y + z\right)\\ \mathbf{elif}\;x \leq 7.4:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, \left(-z\right) - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -5.6e+45)
   (- (* x (log y)) (+ y z))
   (if (<= x 7.4) (- (log t) (+ y z)) (fma (log y) x (- (- z) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -5.6e+45) {
		tmp = (x * log(y)) - (y + z);
	} else if (x <= 7.4) {
		tmp = log(t) - (y + z);
	} else {
		tmp = fma(log(y), x, (-z - y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -5.6e+45)
		tmp = Float64(Float64(x * log(y)) - Float64(y + z));
	elseif (x <= 7.4)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = fma(log(y), x, Float64(Float64(-z) - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -5.6e+45], N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.4], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + N[((-z) - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.5999999999999999e45

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\left(x \cdot \log y - y\right) - z\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \log t + \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \log t + \left(\color{blue}{\left(x \cdot \log y - y\right)} - z\right) \]

      5. associate--l- N/A

         \[\leadsto \log t + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} \]

      6. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} - \left(y + z\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - \left(y + z\right) \]

      11. lower-+.f64 99.6

          \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \color{blue}{\left(y + z\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]

      2. lower-log.f64 99.6

         \[\leadsto x \cdot \color{blue}{\log y} - \left(y + z\right) \]

   7. Applied rewrites 99.6%

      \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]

   if -5.5999999999999999e45 < x < 7.4000000000000004

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log t} - \left(y + z\right) \]

      3. lower-+.f64 99.2

         \[\leadsto \log t - \color{blue}{\left(y + z\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

   if 7.4000000000000004 < x

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} + \log t \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right)} - \left(z - \log t\right) \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\mathsf{neg}\left(y\right)\right)\right)} - \left(z - \log t\right) \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \log y + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      10. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right)} + \log t\right) \]

      13. lower-neg.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(\color{blue}{\left(-y\right)} - z\right) + \log t\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\left(-y\right) - z\right) + \log t\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\log t + \left(\left(\mathsf{neg}\left(y\right)\right) - z\right)}\right) \]

      3. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \log t + \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right)}\right) \]

      4. associate-+r- N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\log t + \left(\mathsf{neg}\left(y\right)\right)\right) - z}\right) \]

      5. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \left(\log t + \color{blue}{\left(\mathsf{neg}\left(y\right)\right)}\right) - z\right) \]

      6. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\log t - y\right)} - z\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\log t - \left(y + z\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \log t - \color{blue}{\left(z + y\right)}\right) \]

      9. associate--r+ N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\log t - z\right) - y}\right) \]

      10. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\log t - z\right) - y}\right) \]

      11. lower--.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\log t - z\right)} - y\right) \]

   6. Applied rewrites 99.8%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\log t - z\right) - y}\right) \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-1 \cdot z} - y\right) \]
   8. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} - y\right) \]

      2. lower-neg.f64 99.5

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(-z\right)} - y\right) \]

   9. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(-z\right)} - y\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y - \left(y + z\right)\\ \mathbf{if}\;x \leq -5.6 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.4:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x (log y)) (+ y z))))
   (if (<= x -5.6e+45) t_1 (if (<= x 7.4) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * log(y)) - (y + z);
	double tmp;
	if (x <= -5.6e+45) {
		tmp = t_1;
	} else if (x <= 7.4) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * log(y)) - (y + z)
    if (x <= (-5.6d+45)) then
        tmp = t_1
    else if (x <= 7.4d0) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * Math.log(y)) - (y + z);
	double tmp;
	if (x <= -5.6e+45) {
		tmp = t_1;
	} else if (x <= 7.4) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * math.log(y)) - (y + z)
	tmp = 0
	if x <= -5.6e+45:
		tmp = t_1
	elif x <= 7.4:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * log(y)) - Float64(y + z))
	tmp = 0.0
	if (x <= -5.6e+45)
		tmp = t_1;
	elseif (x <= 7.4)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * log(y)) - (y + z);
	tmp = 0.0;
	if (x <= -5.6e+45)
		tmp = t_1;
	elseif (x <= 7.4)
		tmp = log(t) - (y + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.6e+45], t$95$1, If[LessEqual[x, 7.4], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.5999999999999999e45 or 7.4000000000000004 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\left(x \cdot \log y - y\right) - z\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \log t + \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \log t + \left(\color{blue}{\left(x \cdot \log y - y\right)} - z\right) \]

      5. associate--l- N/A

         \[\leadsto \log t + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} \]

      6. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} - \left(y + z\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - \left(y + z\right) \]

      11. lower-+.f64 99.7

          \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \color{blue}{\left(y + z\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]

      2. lower-log.f64 99.5

         \[\leadsto x \cdot \color{blue}{\log y} - \left(y + z\right) \]

   7. Applied rewrites 99.5%

      \[\leadsto \color{blue}{x \cdot \log y} - \left(y + z\right) \]

   if -5.5999999999999999e45 < x < 7.4000000000000004

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log t} - \left(y + z\right) \]

      3. lower-+.f64 99.2

         \[\leadsto \log t - \color{blue}{\left(y + z\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{elif}\;x \leq 1350:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log y, x, -z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.2e+121)
   (fma (log y) x (- y))
   (if (<= x 1350.0) (- (log t) (+ y z)) (fma (log y) x (- z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.2e+121) {
		tmp = fma(log(y), x, -y);
	} else if (x <= 1350.0) {
		tmp = log(t) - (y + z);
	} else {
		tmp = fma(log(y), x, -z);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.2e+121)
		tmp = fma(log(y), x, Float64(-y));
	elseif (x <= 1350.0)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = fma(log(y), x, Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.2e+121], N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision], If[LessEqual[x, 1350.0], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], N[(N[Log[y], $MachinePrecision] * x + (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.2000000000000003e121

   1. Initial program 99.5%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} + \log t \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right)} - \left(z - \log t\right) \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\mathsf{neg}\left(y\right)\right)\right)} - \left(z - \log t\right) \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \log y + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      10. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right)} + \log t\right) \]

      13. lower-neg.f64 99.4

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(\color{blue}{\left(-y\right)} - z\right) + \log t\right) \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\left(-y\right) - z\right) + \log t\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-1 \cdot y}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\mathsf{neg}\left(y\right)}\right) \]

      2. lower-neg.f64 93.8

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-y}\right) \]

   7. Applied rewrites 93.8%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-y}\right) \]

   if -4.2000000000000003e121 < x < 1350

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log t} - \left(y + z\right) \]

      3. lower-+.f64 95.9

         \[\leadsto \log t - \color{blue}{\left(y + z\right)} \]

   5. Applied rewrites 95.9%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

   if 1350 < x

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} + \log t \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right)} - \left(z - \log t\right) \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\mathsf{neg}\left(y\right)\right)\right)} - \left(z - \log t\right) \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \log y + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      10. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right)} + \log t\right) \]

      13. lower-neg.f64 99.8

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(\color{blue}{\left(-y\right)} - z\right) + \log t\right) \]

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\left(-y\right) - z\right) + \log t\right)} \]

   5. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-1 \cdot z}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\mathsf{neg}\left(z\right)}\right) \]

      2. lower-neg.f64 85.1

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-z}\right) \]

   7. Applied rewrites 85.1%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-z}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\log y, x, -y\right)\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 8.8 \cdot 10^{+18}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (log y) x (- y))))
   (if (<= x -4.2e+121) t_1 (if (<= x 8.8e+18) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(log(y), x, -y);
	double tmp;
	if (x <= -4.2e+121) {
		tmp = t_1;
	} else if (x <= 8.8e+18) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(log(y), x, Float64(-y))
	tmp = 0.0
	if (x <= -4.2e+121)
		tmp = t_1;
	elseif (x <= 8.8e+18)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[y], $MachinePrecision] * x + (-y)), $MachinePrecision]}, If[LessEqual[x, -4.2e+121], t$95$1, If[LessEqual[x, 8.8e+18], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.2000000000000003e121 or 8.8e18 < x

   1. Initial program 99.7%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} + \log t \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right) - \left(z - \log t\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y - y\right)} - \left(z - \log t\right) \]

      5. sub-neg N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \left(\mathsf{neg}\left(y\right)\right)\right)} - \left(z - \log t\right) \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{x \cdot \log y + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \color{blue}{\log y \cdot x} + \left(\left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\mathsf{neg}\left(y\right)\right) - \left(z - \log t\right)\right)} \]

      10. associate--r- N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      11. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right) + \log t}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) - z\right)} + \log t\right) \]

      13. lower-neg.f64 99.7

          \[\leadsto \mathsf{fma}\left(\log y, x, \left(\color{blue}{\left(-y\right)} - z\right) + \log t\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\log y, x, \left(\left(-y\right) - z\right) + \log t\right)} \]

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-1 \cdot y}\right) \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{\mathsf{neg}\left(y\right)}\right) \]

      2. lower-neg.f64 87.7

         \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-y}\right) \]

   7. Applied rewrites 87.7%

      \[\leadsto \mathsf{fma}\left(\log y, x, \color{blue}{-y}\right) \]

   if -4.2000000000000003e121 < x < 8.8e18

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log t} - \left(y + z\right) \]

      3. lower-+.f64 96.0

         \[\leadsto \log t - \color{blue}{\left(y + z\right)} \]

   5. Applied rewrites 96.0%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 84.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \log y\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{+121}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+83}:\\ \;\;\;\;\log t - \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (log y))))
   (if (<= x -4.5e+121) t_1 (if (<= x 5.5e+83) (- (log t) (+ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * log(y);
	double tmp;
	if (x <= -4.5e+121) {
		tmp = t_1;
	} else if (x <= 5.5e+83) {
		tmp = log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * log(y)
    if (x <= (-4.5d+121)) then
        tmp = t_1
    else if (x <= 5.5d+83) then
        tmp = log(t) - (y + z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * Math.log(y);
	double tmp;
	if (x <= -4.5e+121) {
		tmp = t_1;
	} else if (x <= 5.5e+83) {
		tmp = Math.log(t) - (y + z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * math.log(y)
	tmp = 0
	if x <= -4.5e+121:
		tmp = t_1
	elif x <= 5.5e+83:
		tmp = math.log(t) - (y + z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * log(y))
	tmp = 0.0
	if (x <= -4.5e+121)
		tmp = t_1;
	elseif (x <= 5.5e+83)
		tmp = Float64(log(t) - Float64(y + z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * log(y);
	tmp = 0.0;
	if (x <= -4.5e+121)
		tmp = t_1;
	elseif (x <= 5.5e+83)
		tmp = log(t) - (y + z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[Log[y], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e+121], t$95$1, If[LessEqual[x, 5.5e+83], N[(N[Log[t], $MachinePrecision] - N[(y + z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.5000000000000003e121 or 5.4999999999999996e83 < x

   1. Initial program 99.6%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \log y} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{x \cdot \log y} \]

      2. lower-log.f64 75.3

         \[\leadsto x \cdot \color{blue}{\log y} \]

   5. Applied rewrites 75.3%

      \[\leadsto \color{blue}{x \cdot \log y} \]

   if -4.5000000000000003e121 < x < 5.4999999999999996e83

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log t} - \left(y + z\right) \]

      3. lower-+.f64 94.4

         \[\leadsto \log t - \color{blue}{\left(y + z\right)} \]

   5. Applied rewrites 94.4%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{-y}{z}, -z\right)\\ \mathbf{if}\;z \leq -6650:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 330:\\ \;\;\;\;\log t - y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ (- y) z) (- z))))
   (if (<= z -6650.0) t_1 (if (<= z 330.0) (- (log t) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (-y / z), -z);
	double tmp;
	if (z <= -6650.0) {
		tmp = t_1;
	} else if (z <= 330.0) {
		tmp = log(t) - y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(Float64(-y) / z), Float64(-z))
	tmp = 0.0
	if (z <= -6650.0)
		tmp = t_1;
	elseif (z <= 330.0)
		tmp = Float64(log(t) - y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[((-y) / z), $MachinePrecision] + (-z)), $MachinePrecision]}, If[LessEqual[z, -6650.0], t$95$1, If[LessEqual[z, 330.0], N[(N[Log[t], $MachinePrecision] - y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6650 or 330 < z

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\left(x \cdot \log y - y\right) - z\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \log t + \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \log t + \left(\color{blue}{\left(x \cdot \log y - y\right)} - z\right) \]

      5. associate--l- N/A

         \[\leadsto \log t + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} \]

      6. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} - \left(y + z\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - \left(y + z\right) \]

      11. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \color{blue}{\left(y + z\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z} + 1\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)\right)} + \left(\mathsf{neg}\left(z\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \color{blue}{z \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}} + \left(\mathsf{neg}\left(z\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\left(\log t + x \cdot \log y\right) - y}{z}, \mathsf{neg}\left(z\right)\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(x, \log y, \log t\right) - y}{z}, -z\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{-1 \cdot y}{z}, \mathsf{neg}\left(z\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.3%

       \[\leadsto \mathsf{fma}\left(z, \frac{-y}{z}, -z\right) \]

   if -6650 < z < 330

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]
   4. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

      2. lower-log.f64 N/A

         \[\leadsto \color{blue}{\log t} - \left(y + z\right) \]

      3. lower-+.f64 67.5

         \[\leadsto \log t - \color{blue}{\left(y + z\right)} \]

   5. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\log t - \left(y + z\right)} \]

   6. Taylor expanded in z around 0

      \[\leadsto \log t - \color{blue}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.5%

      \[\leadsto \log t - \color{blue}{y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{-y}{z}, -z\right)\\ \mathbf{if}\;z \leq -6650:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 120000000:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (/ (- y) z) (- z))))
   (if (<= z -6650.0) t_1 (if (<= z 120000000.0) (- y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (-y / z), -z);
	double tmp;
	if (z <= -6650.0) {
		tmp = t_1;
	} else if (z <= 120000000.0) {
		tmp = -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(Float64(-y) / z), Float64(-z))
	tmp = 0.0
	if (z <= -6650.0)
		tmp = t_1;
	elseif (z <= 120000000.0)
		tmp = Float64(-y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[((-y) / z), $MachinePrecision] + (-z)), $MachinePrecision]}, If[LessEqual[z, -6650.0], t$95$1, If[LessEqual[z, 120000000.0], (-y), t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6650 or 1.2e8 < z

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right) + \log t} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\log t + \left(\left(x \cdot \log y - y\right) - z\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \log t + \color{blue}{\left(\left(x \cdot \log y - y\right) - z\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \log t + \left(\color{blue}{\left(x \cdot \log y - y\right)} - z\right) \]

      5. associate--l- N/A

         \[\leadsto \log t + \color{blue}{\left(x \cdot \log y - \left(y + z\right)\right)} \]

      6. associate-+r- N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\log t + x \cdot \log y\right) - \left(y + z\right)} \]

      8. +-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \log y + \log t\right)} - \left(y + z\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \log y} + \log t\right) - \left(y + z\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right)} - \left(y + z\right) \]

      11. lower-+.f64 99.9

          \[\leadsto \mathsf{fma}\left(x, \log y, \log t\right) - \color{blue}{\left(y + z\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \log y, \log t\right) - \left(y + z\right)} \]

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot z\right) \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \cdot \left(1 + -1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) \]

      3. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \color{blue}{\left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z} + 1\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) + \left(\mathsf{neg}\left(z\right)\right) \cdot 1} \]

      5. *-rgt-identity N/A

         \[\leadsto \left(\mathsf{neg}\left(z\right)\right) \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right) + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(z \cdot \left(-1 \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)\right)} + \left(\mathsf{neg}\left(z\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(z \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}\right)\right)}\right)\right) + \left(\mathsf{neg}\left(z\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \color{blue}{z \cdot \frac{\left(\log t + x \cdot \log y\right) - y}{z}} + \left(\mathsf{neg}\left(z\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\left(\log t + x \cdot \log y\right) - y}{z}, \mathsf{neg}\left(z\right)\right)} \]

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{\mathsf{fma}\left(x, \log y, \log t\right) - y}{z}, -z\right)} \]

   8. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(z, \frac{-1 \cdot y}{z}, \mathsf{neg}\left(z\right)\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 72.8%

       \[\leadsto \mathsf{fma}\left(z, \frac{-y}{z}, -z\right) \]

   if -6650 < z < 1.2e8

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 41.9

         \[\leadsto \color{blue}{-y} \]

   5. Applied rewrites 41.9%

      \[\leadsto \color{blue}{-y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 46.8% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+130}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+93}:\\ \;\;\;\;-y\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.3e+130) (- z) (if (<= z 4.8e+93) (- y) (- z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+130) {
		tmp = -z;
	} else if (z <= 4.8e+93) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.3d+130)) then
        tmp = -z
    else if (z <= 4.8d+93) then
        tmp = -y
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e+130) {
		tmp = -z;
	} else if (z <= 4.8e+93) {
		tmp = -y;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.3e+130:
		tmp = -z
	elif z <= 4.8e+93:
		tmp = -y
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.3e+130)
		tmp = Float64(-z);
	elseif (z <= 4.8e+93)
		tmp = Float64(-y);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.3e+130)
		tmp = -z;
	elseif (z <= 4.8e+93)
		tmp = -y;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.3e+130], (-z), If[LessEqual[z, 4.8e+93], (-y), (-z)]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e130 or 4.80000000000000021e93 < z

   1. Initial program 99.9%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot z} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 76.6

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 76.6%

      \[\leadsto \color{blue}{-z} \]

   if -1.2999999999999999e130 < z < 4.80000000000000021e93

   1. Initial program 99.8%

      \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{-1 \cdot y} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

      2. lower-neg.f64 41.2

         \[\leadsto \color{blue}{-y} \]

   5. Applied rewrites 41.2%

      \[\leadsto \color{blue}{-y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 30.1% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ -y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- y))

C

double code(double x, double y, double z, double t) {
	return -y;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -y
end function

Java

public static double code(double x, double y, double z, double t) {
	return -y;
}

Python

def code(x, y, z, t):
	return -y

Julia

function code(x, y, z, t)
	return Float64(-y)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -y;
end

Wolfram

code[x_, y_, z_, t_] := (-y)

Derivation

1. Initial program 99.8%

   \[\left(\left(x \cdot \log y - y\right) - z\right) + \log t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{-1 \cdot y} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

   2. lower-neg.f64 29.5

      \[\leadsto \color{blue}{-y} \]

5. Applied rewrites 29.5%

   \[\leadsto \color{blue}{-y} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:incompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (- (* x (log y)) y) z) (log t)))